Uniformly continuous function - modulus of continuity

Question

Give an example of a uniformly continuous function $f: (X,d) \rightarrow (Y,\rho)$ for which there doesn't exists a modulus of continuity $\omega: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that:

$$\forall x,y \in X: \ \ \rho(f(x), f(y)) \le \omega(d(x,y)).$$

I cannot come up with anything. Could you help me with that?

Thank you.

Answer 1

It is easy to come up with many examples if you keep in mind that

• if $d$ is the discrete metric, $f$ is already uniformly continuous;
• if the modulus of continuity is finite everywhere, then the image of every bounded set is also bounded.